edit: If you have the space, a Nintendo Wii running Genesis Plus GX with the video mode set to "Original" and a CRT with at least composite (yellow cable) blows all of those handhelds out of the water. If you only have composite, the emulator has a "trap filter" option which utilizes some custom hardware and the Nintendo Wii's graphics chip to isolate the chroma and luma channels, resulting in a much cleaner composite image. Of course, component is still better, but it's becoming harder to find a 15kHz CRT with component hookups.

Otherwise, I recommend Retroid Pocket 6 if you like Android or want to play GC/PS2/N64, or a Miyoo Mini Plus if you don't care about any of that. Neat thing about the mini plus is its 640x480 screen, so you get Genesis/Mega Drive games edge to edge 2x integer scale if configured properly. Also has OnionOS which is a pretty good custom firmware.

Other than that, lots of these things have major problems. Watch out for things like improperly calibrated joysticks (can't run in certain directions in 3D platformers, only walking speed), screen tearing, saves disappearing from the filesystem if the system shuts down improperly during gameplay, audio crackle/poor audio in general, IPS bleed around display edges, soggy dpads, face buttons too firm, etc. A lot of problems with any given device you can run into that don't fall under the usual umbrella of review metrics, so you won't know until you realize a week into owning it why something feels off about it...

Incidentally, there are remakes of the first two for PS2, released under the Sega AGES line of discount games. Unfortunately, only in Japanese at the time I played, though translation patches are now available. They’re called Phantasy Star Generation 1 and 2. There is also a release of a compilation of the original games for PS2.

PS = Phantasy Star